Elegant labeling of the web graph without center W(2, n) and W(3, n)

Let G be a finite graph without multiple edges nor loops, with vertex set V(G),edge setE(G),and number of edgesq. An elegant labelingfof graph G is a function f: V(G) → {0, 1, 2, …, q}, where f is injective, such that the induced edge label f*(xy) = f(x) + f(y) (mod(q + 1)), where xy ∈ E(G), are distinct and non zero. A graph is said to be elegant if there exists an elegant labeling of the graph.Many authors have studied elegant labeling of graphs and many elegant graphs have been found.There are many graphs are not known whether they are elegant or not. We need to find some new classes of graphs that are elegant. In this paper we show that for every odd integer n, n ≥ 3, the web graphs without center W₀(2,n) and W₀(3,n) are elegant.